Regularization of Second Order Sliding Mode Control Systems

This behaviour, called approximability in [18], [19] and regularization in [17], has been analysed for sliding mode control of first order and must be regarded as a basic procedure in the

Control Systems

Giorgio Bartolini1, Elisabetta Punta2, and Tullio Zolezzi3

1 Department of Electrical and Electronic Engineering, University of Cagliari, Piazza d’Armi - 09123 Cagliari, Italy

giob@dist.unige.it

2

Institute of Intelligent Systems for Automation, National Research Council of Italy (ISSIA-CNR), Via De Marini, 6 - 16149 Genoa, Italy

punta@ge.issia.cnr.it

3

Department of Mathematics, University of Genoa, Via Dodecaneso, 35 - 16146 Genoa, Italy

zolezzi@dima.unige.it

The numerical representation of singular systems [1], with index greater than two and inconsistent initial conditions, presents common features with the implementation of higher order sliding motions, [2], [3] Indeed higher order sliding modes can be viewed as a way to achieve constrained motions, often expressible as an output-zeroing problem after a transient of finite duration [4] The choice of the sliding output is the first step of a sliding mode design process (e.g invariance [5]) If the actual control affects the time derivative

of the sliding output, starting from a certain order k ≥ 1, the corresponding

constrained motion, if attainable, is said to be a k-th order sliding motion [6],

[7], [8], [9], [10] The notion of sliding order is equivalent to the one of relative degree [4]

Discontinuity of the control naturally arises when uncertainties affect the mathematical description of the system be controlled Indeed the reduction to zero in finite time of the sliding variable implies the stepwise solution of differen-tial inequalities of order greater than one with the aim of ensuring a contractive behaviour, in literature referred to as the Fuller phenomenon [11], [12] For higher order sliding mode the procedure is much more involved than in the first order case, for which the application of the comparison lemma, [13], suffices, and a variety of algorithms can be identified

In the literature most of the results are related to the development of second order sliding mode algorithms [6], [7], [14], [8], [15], [9]



Work partially supported by MURST, Progetto Cofinanziato “Controllo, ottimiz-zazione e stabilit`a di sistemi non lineari”, by PRAI-FESR Liguria and by MUR-FAR Project n 630

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 3–21, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

The finite time transient, the reaching phase, is not at issue in this chap-ter, which is devoted to analyze the motion constrained on the sliding surface, which is ideally represented by differential equations with discontinuous r.h.s., the solution of which is defined according to Filippov’s theory, [16], [17] The aim of this chapter is that of providing a theoretical framework to analyse the behaviour of the systems controlled by second order sliding mode when, due

to non-idealities of unspecified nature, the chosen constraints are only approxi-matively satisfied

A class of regular perturbations needs to be identified, so that all the cor-responding real state trajectories, which fulfill only approximately the sliding conditions due to non-idealities of any nature, converge to the unique solution (if it exists) of the differential algebraic equations representing the exact fulfill-ment of the chosen constraints This behaviour, called approximability in [18], [19] and regularization in [17], has been analysed for sliding mode control of first order and must be regarded as a basic procedure in the mathematical analysis and validation of sliding mode control techniques

The given system fulfills the approximability property whenever all real states converge uniformly (on the fixed bounded time interval we consider) to the ideal state as the non-idealities disappear Such a property is of interest since it guar-antees that the dynamical behavior of the real states, we obtain in practice, are close to that of the ideal sliding state, which is the main goal of the sliding mode control techniques This property, discussed in [17], was mathematically formal-ized in [18] and [20] for nonlinear control systems There, sufficient conditions for approximability were found, generalizing the regularization results of [17], about first order sliding mode control methods

In this chapter the approximability results about first order sliding mode control methods are presented The analysis is developed for sliding mode control

of order two, thus obtaining second order regularization results of sliding mode control systems A major result we obtain is that first order implies second order approximability (Theorem 4, Section 4) under mild assumptions Hence the regularization of second order methods is automatically valid as soon as it

is guaranteed for first order sliding techniques

Results on first order approximability are summarized in Section 2 In Sec-tion 3 we introduce a new framework to define rigorously the approximability property for systems, the ideal solution of which is achieved by second order sliding mode methods We compare second order approximability with the one related to first order methods In Section 4 we prove sufficient conditions for second order approximability In Section 5 we obtain some sliding error esti-mate In Section 6 we extend the previous approach to non-idealities occurring

on either the sliding output or the control law Suitable examples are proposed throughout this chapter

A different approach to approximability properties, based on optimization techniques, is considered in [19] and [21] for first and second order sliding methods

2 First Order Approximability: Definitions and Results

We consider sliding mode control problems

˙x = f (t, x, u), u ∈ U;

s(t, x) = 0, 0≤ t ≤ T, x(0) = x0∈ Z, (1)

on a fixed bounded time interval [0, T ], where x ∈ IR N , u ∈ U, U is a closed

subset of IR K , and s ∈ IR M , T > 0 We assume f ∈ C2([0, T ] × IR N × W ) with

W an open set containing U , Z ⊂ IR N , and s ∈ C2([0, T ] × IR N)

Following the terminology of [17], page 13, and the approach of [18], [20], we

deal with ideal states which fulfill exactly the sliding condition s[t, x(t)] = 0 for every t, as opposite to real states, fulfilling only approximately such a condition.

A parameter ε, belonging to a metric space with a fixed element conventionally

denoted by 0, will be used to represent non-idealities of any nature in the real

sliding We write xε → x to denote convergence of x ε k towards x for each sequence εk → 0 The notation x ε − → →x means uniform convergence on [0, T ].

We denote by U ∞ the set of all admissible control laws in (1), defined as follows These are all functions

u : [0, T ] × IR N → U

which are L ⊗ B − measurable, i.e measurable with respect to the σ − algebra

generated by the products of the Lebesgue measurable subset of [0, T ] and the Borel measurable subset of IR N, and which fulfill the following property For any

such u, the differential system in (1) has an almost everywhere or a Filippov solution x on [0, T ] such that u[ ·, x(·)] ∈ L ∞ (0, T ).

To simplify the notations, we shall write

˙

x = f (t, x, u) on [0, T ] meaning that u ∈ U ∞ and x is either an almost everywhere or a Filippov solution

on [0, T ].

The sliding mode control system (1) fulfills the first order approximability property whenever the following is true For every x0∈ Z such that s(0, x0) = 0

there exists a unique sliding state y, i.e for some control u ∗ ∈ U ∞, not necessarily unique, we have

˙

y = f (t, y, u ∗) on [0, T ], y(0) = x

s[t, y(t)] = 0, 0≤ t ≤ T. (3)

Moreover for every sequence (u ε , x ε ) such that ˙x ε = f (t, x ε , u ε ) on [0, T ] and

s(t, x ε )− → →0 we have x ε − → →y provided x ε(0)→ y(0).

The above definition, compared with that presented in [18] where Z = IR N,

does not require either uniqueness of the sliding control law u ∗ or existence of the equivalent control, moreover s is allowed to depend on t as well.

Often the sliding manifold is reached at points of a restricted part of it, thus

the constraint we introduce on the initial states by the set Z.

Uniqueness of the sliding state is fulfilled, under standard assumptions, when-ever the equivalent control is available, see [17], [18]

To be more specific, we consider the first order ideal system made up of control-state pairs (v, y) such that



˙

y = f (t, y, v) on [0, T ], v ∈ U ∞;

˙s = st(t, y) + sx(t, y)f (t, y, v) = 0.

The first order real control-state pairs are given by pairs (u ε , x ε ), where x ε is

absolutely continuous in [0, T ], and such that for almost every t ∈ (0, T )



˙

x ε = f (t, xε , u ε), u ε(t) ∈ U;

s t(t, xε) + sx(t, xε)f (t, xε , u ε) = mε(t). (4) About the non-idealities mεwe shall employ the condition

m ε → 0 in W −1,∞ (0, T ) as ε → 0, (5)

which means that mε ∈ L1(0, T ) and sup {|t

0m ε(r)dr | : 0 ≤ t ≤ T } → 0 as

ε → 0.

Here W −1,∞ (0, T ) denotes, as usual, the dual space of the Sobolev space

W 1,1 (0, T ) (see e.g [22])

The following conditions will be referred to in the sequel

For every compact set S ⊂ U there exist A1, B1∈ L1(0, T ) such that

|f(t, x, u)| ≤ A1(t) |x| + B1(t) (6)

for a.e t, every u ∈ S and x ∈ IR N

For every compact set Z ⊂ IR N × U there exists C1∈ L1(0, T ) such that

|f(t, x  , u) − f(t, x  , u) | ≤ C1(t) |x  − x  | (7)

for a.e t, every (x  , u) and (x  , u) ∈ Z.

Theorem 1 The control system (1) fulfills the first order approximability

prop-erty if conditions (6) and (7) are met, U is compact, f (t, x, U ) is convex for a.e t and every x with s(t, x) = 0, and for every x0∈ Z with s(t, x0) = 0 there exists

a unique sliding state issued from x0

For the proof see [19]

The convexity condition on f , which is required by Theorem 1, can be relaxed

in some significant cases according to the following theorems

Theorem 2 Let the control system (1) be such that

f (t, x, u) = A(x) + B(x)h(u).

The first order approximability property holds provided:

• U is compact, h(U) is convex;

• for every K > 0 there exist constants C2, D2 such that|s(x)| ≤ K implies

|A(x)| + |B(x)| ≤ C2|x| + D2;

• s x(x)B(x) = G(x) is nonsingular near the sliding manifold and we have

uniqueness in the large for every initial value problem

˙z = A(z) + B(z)ν(z), z(0) given with s [z(0)] ,

where ν(z) = −G −1 (z)sz(z)A(z).

For the proof see [20]

Theorem 3 Let the control system (1) be such that

f (t, x, u) = [x2, x3, , x N , g(x, u)] T ,

where u ∈ IR.

The first order approximability property holds provided:

• U is compact;

• for every K > 0 there exist constants A2, B2 such that|s(x)| ≤ K implies

|g(x, u)| ≤ A2|x| + B2;

• we have uniqueness in the large for every initial value problem

˙

x i = xi+1 , i = 1, , N − 1,

˙

x n ∂s

∂x n +N−1

j=1 x j+1 ∂s

∂x j = 0, s [x(0)] = 0.

For the proof see [20]

Recently, in [23], the approximability property has been extended to systems

in regular form, which do not necessarily satisfy the convexity condition Let the control system (1) be such that

f (t, x, u) = (f1(x) T , f2(x, u) T)T ,

where x = (x T

1, x T

2)T ∈ IR N , x1∈ IR N −K , x

2∈ IR K , u ∈ U ⊂ IR K , U compact,

f1: IR N → IR N −M and f

2: IR N × U → IR M Consider

s(x) = x2− h(x1), where s : IR N → IR M and h : IR N −M → IR M is a C1 function

The first order approximability property holds provided

h x1(x1)f1(x1, h(x1))∈ cof2(x1, h(x1), U ).

The class of systems for which the approximability holds, includes then any

set of k coupled differential equations

z (n i)

i = f i (z, u), i = 1, , k,

with z =



z1, , z (n1−1) , , z

k , , z (n1−1)T

and u = (u1, , u k) T

3 Second Order Approximability: Definitions

The second order ideal system of control-state pairs (u, z), both absolutely con-tinuous in [0, T ], such that for almost every t ∈ (0, T )



˙z = f (t, z, u), u(t) ∈ U;

¨

s = P (t, z, u) + Q(t, z, u) ˙u = 0, (8)

where

P = s tt + 2stx f + f T s xx f + s x f t + sx f x f,

Q = s x f u and f T s xx f denotes the vector of components f T s jxx f , j = 1, , M

We model the non-idealities acting on the second order ideal system (8) by

using two different terms The first, denoted by b ε = b ε (t) ∈ IR M, takes into ac-count second order sliding non-idealities, so that in the real second order system

we have ¨s = b ε The second, denoted by c ε = c ε (t) ∈ IR K, takes into account non-idealities in obtaining ˙u ε, so that in the real second order system we work with wε = ˙uε + cε instead of wε = ˙uε.

Specific properties of the non-idealities acting on the second order system may depend on the particular control problem at hand Therefore we fix a nonempty

subset N0 ⊂ L1(0, T ) of sequences (bε , c ε), and we consider non-idealities be-longing to N0

The second order real control-state pairs are thereby given by pairs (uε , x ε), both absolutely continuous in [0, T ], such that for almost every t ∈ (0, T )

⎧

⎨

⎩

˙

x ε = f (t, x ε , u ε ), u ε (t) ∈ U;

P (t, x ε , u ε ) + Q(t, x ε , u ε )w ε = b ε (t);

w ε (t) = ˙ u ε (t) + c ε (t).

(9)

About the non-idealities c εwe shall employ the condition

c ε → 0 in W −1,∞ (0, T ) as ε → 0, (10)

which means that cε ∈ L1(0, T ) and sup {|t

0c ε(r)dr | : 0 ≤ t ≤ T } → 0 as

ε → 0.

About bε we consider two different ways bε can vanish as ε → 0, namely

b ε → 0 in W −2,∞ (0, T ), (11)

either

b ε → 0 in W −1,∞ (0, T ). (12)

By (11) we mean that b ε ∈ L1(0, T ) and sup {|θ ε (t) | : 0 ≤ t ≤ T } → 0 where

¨

ε = b ε almost everywhere in (0, T ), θ ε (0) = ˙θ ε(0) = 0

Here W −1,∞ (0, T ) and W −2,∞ (0, T ) denote, as usual, the dual spaces of the Sobolev spaces W 1,1 (0, T ) and W 2,1 (0, T ) respectively (see e.g [22]).

Accordingly, we formulate two definitions of second order approximability of

(1) within N Both second order approximability properties identify second

order sliding mode control systems such that real states are uniformly close to the unique sliding state, as the non-idealities acting on the system are suitably small Roughly speaking, first kind approximability means that such a robust

behaviour of the control system is guaranteed whenever ˙s is uniformly small.

Second kind approximability means that the same behaviour is present whenever

s is small This property is similar to that required in the definition of first order

approximability (by the quite different first order methods) In each definition

we require the following condition

Condition A: For every x0 ∈ Z such that s(0, x0) = 0 there exists a unique

sliding state y corresponding to some continuous control u ∗, i.e (2), (3) are

fulfilled

Definition 1 Second order approximability of the first kind within N0 means the following First, condition A is fulfilled Second, given any sequences b ε , c ε

in N0 we require that x ε − → →y provided x ε(0)→ y (0), u ε(0) → u ∗ (0) for every

sequences x ε satisfying (9) such that (10) and (12) are fulfilled.

Definition 2 Second order approximability of the second kind within N0means the same as in Definition 1, except that (12) is replaced by (11).

Since strong convergence in W −1,∞ (0, T ) implies the same in W −2,∞ (0, T ), we

have that second kind implies first kind approximability

To define N0, we shall consider the following conditions about sequences bε and cε:

supε

T

0 (|b ε (t) | + |c ε (t) |)dt < +∞, (13)

supε

T

0

| ˙Q[t, x ε(t), uε(t)] |dt < +∞. (14) Condition (13) is needed here for technical reasons (however see Example 3)

Condition (14) (again needed for technical reasons) is fulfilled provided U is

compact and

|f(t, x, u)| ≤ a(t) + b(t)|x|

for almost every t, all x and u, with a, b ∈ L1(0, T ) This follows in a standard

way by Gronwall’s lemma

We emphasize that the approximability properties we have defined are in-dependent of the particular algorithm used to enforce any second order sliding motion

Let N0 be defined by the property that supε |b ε | ∈ L1(0, T ), a stronger bound-edness condition on bε than (13); then the two corresponding approximability properties are indeed equivalent, as a corollary of the following proposition

Proposition 1 Let supε |b ε | ∈ L1(0, T ) Then convergence of bε in W −2,∞ (0, T ) implies convergence in W −1,∞ (0, T ).

For the proof see [24].

The main result of this section is the following

Theorem 4 Let N0be defined by (13) and (14) If system (1) fulfills first order

approximability and, for all x0∈ Z with s(0, x0) = 0, the corresponding sliding state can be generated by a continuous control, then (1) fulfills both second order

approximability properties within N0

For the proof see [24]

Continuity of a sliding control is not a very restrictive assumption, being fulfilled whenever the equivalent control is available, as it often happens in sliding mode control applications, see [17]

According to Theorem 4, if first order sliding mode control methods cannot give rise to ambiguous motions, no further ambiguous behaviour can be induced

by any second order control algorithm Hence the validation of second order methods relies on checking known criteria yielding first order approximability,

as those known from [18], [20] and [19]

Remark: A slightly weaker definition of first and second order approximability

properties can be given by requiring that xε − → →y for all the sequences (u ε , x ε) of

real control-state pairs such that sup u ε ∞ < + ∞, where · ∞ denotes the

L ∞ (0, T ) norm Thus we require convergence to the ideal state y only when

the control laws are uniformly bounded Then the proof of Theorem 4 shows that, with these weaker definitions, first order approximability still implies both second order corresponding properties

The same proof (see [24]) shows that second order real states of (1) are also first order real states The next example shows that the converse to Theorem 4 fails A sliding mode control system may fulfill second kind approximability of second order, and may fail to possess first order approximability

Example 1 Consider

˙

x1= u1, x˙2= u2, x˙3= u1u2, 0≤ t ≤ 1,

|u1| ≤ 1, |u2| ≤ 1, s1(x) = x1, s2(x) = x2,

with Z = IR3

The only sliding control is the equivalent control u ∗= 0 and the only sliding state, given y(0), is the constant y(t) = y(0) This system does not fulfill first

order approximability, see [18]

Let us show that second kind approximability of second order is however

fulfilled within N0defined by (13) and (14)

Given b ε , c ε as in (11), (10) respectively, and satisfying (13), let x ε(0)→ y(0),

u ε(0)→ 0 Since P = 0 and Q =

1 0

0 1

we have

x jε(t) = xjε(0) +t

0u jε dr, j = 1, 2,

u ε(t) = uε(0) +t

0b ε dr −t

0c ε dr.

0u ε dr− → →0, whence x jε (t)− → →y j(0) We compute

x 3ε (t) − x 3ε(0) =t

0u 1ε u 2ε dr

= u 2ε (t)t

0u 1ε dr −t

0

r

0 u 1ε dα

(b 2ε − c 2ε ) dr.

We have u 2ε (t)t

0u 1ε dr− → →0, and remembering (13) we get



t

0

r

0 u 1ε dα

(b 2ε − c 2ε ) dr

≤T

0 (|b 2ε | + |c 2ε |) dr max rr

0 u 1ε dα

≤ (constant) max rr

0 u 1ε dα  → 0.

It follows that x 3ε − → →y3(0) hence x ε − → →y, yielding second order approximability.

The following example presents a case of applicability of the second order ap-proximability

Example 2 Let y d (t) ∈ C2([0, T ]) be an available signal such that |¨y d (t) | ≤ L

for every t Consider the sliding mode control system

˙

x1= x2, x˙2= u, |u| ≤ L;

s(t, x) = x1− y d (t) , 0≤ t ≤ T. (15)

Here N = 2 and K = M = 1 Since Q = s x f u= 0 everywhere, the approximabil-ity criteria developed in [18] do not apply We check first order approximabilapproximabil-ity via Corollary 4.1 of [19] The required properties of linear growth and local Lipschitz continuity of the dynamics are obviously fulfilled, as well as

convex-ity of f (t, x, U ) Given x0 ∈ IR2 such that s(t, x0) = 0, i.e x01(0) = yd(0),

if x02(0) = ˙y d(0) there exists a unique sliding state y issued from x0, namely

y = (y d , ˙ y d) T , which corresponds to the continuous control u = ¨ y d(in the almost

everywhere sense) Hence, by Theorem 4 with Z =



(y d (0) , ˙ y d(0))T

 , second

order approximability holds Then x 1ε copies yd and x 2εcopies ˙y d This happens for every non-idealities bε, cεacting on the system and fulfilling (10), either (11)

or (12), (13), (14), independently of the particular second order sliding mode algorithm employed to control the system

Let us consider system (15) in the second order mode (9) affected by the

non-ideality bε = sin ε t

, cε = 0, ε > 0 Since the initial values converge and both

s ε and ˙sε converge uniformly to 0, as ε → 0, if the actual ε is small enough, x 1ε copies yd and x 2εcopies ˙y d (Figures 1 and 2)

The following example shows the possible implications of the failure of condition (13) for the second order approximability

Example 3 We consider the same sliding mode control system of Example 2,

namely

˙x1= x2, ˙x2= u; s(t, x) = x1− y d (t) , 0≤ t ≤ T,

except the constraint|u| ≤ L, with non-idealities given by

0 5 10 15

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t [sec]

x 1 ε

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

t [sec]

2π)

−3

−2

−1

0

1

2

3

t [sec]

x 2 ε

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

t [sec]

2π)

b ε(t) = 1

εcos

t ε

, c ε(t) = 0, ε > 0, 0≤ t ≤ T.

Here the control region is the whole real line, bε → 0 in W −2,∞ (0, T ), bε

W −1,∞ (0, T ) Let (uε , x ε) be any second order real control-state pair such that

x 1ε(0)→ y d(0), x 2ε(0)→ ˙y d(0) Then we have

˙sε(t) = ˙sε(0) + sin t ε

, ˙s ε(0) = x 2ε(0)− ˙y d(0),

s ε(t) = t ˙sε(0) + ε

1− cos t

ε



.

Since the initial values converge and s ε converges uniformly to 0, as ε → 0, if

the actual ε is small enough, x 1ε copies y d (Figure 3) However, since ˙s ε does

not converge uniformly to 0, as ε → 0, not even point-wise, then, even if the

actual ε is small, x 2ε= ˙x 1ε does not copy ˙y d (Figure 4) Here the point is that unbounded non-idealities are allowed and condition (13) fails In such a case first order approximability may fail, even if, as checked in Example 2, the same property is present, provided the control region is bounded Indeed consider the

first order real control-state pairs (uε , x ε) such that

Let us consider system (15) in the second order mode (9) affected by the

non-ideality... shows the possible implications of the failure of condition (13) for the second order approximability

Example We consider the same sliding mode control system of Example 2,

namely... x ε − → →y, yielding second order approximability.

The following example presents a case of applicability of the second order ap-proximability

Example